Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 24.1s

Precision: 64

Math

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

↓

\[\left(b \cdot b\right) \cdot \left(4 - a \cdot 12\right) + \left(4 \cdot \left(a \cdot a\right) + \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot a + \left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right)\right)\right)\]

C

double f(double a, double b) {
        double r9172767 = a;
        double r9172768 = r9172767 * r9172767;
        double r9172769 = b;
        double r9172770 = r9172769 * r9172769;
        double r9172771 = r9172768 + r9172770;
        double r9172772 = 2.0;
        double r9172773 = pow(r9172771, r9172772);
        double r9172774 = 4.0;
        double r9172775 = 1.0;
        double r9172776 = r9172775 + r9172767;
        double r9172777 = r9172768 * r9172776;
        double r9172778 = 3.0;
        double r9172779 = r9172778 * r9172767;
        double r9172780 = r9172775 - r9172779;
        double r9172781 = r9172770 * r9172780;
        double r9172782 = r9172777 + r9172781;
        double r9172783 = r9172774 * r9172782;
        double r9172784 = r9172773 + r9172783;
        double r9172785 = r9172784 - r9172775;
        return r9172785;
}

↓

double f(double a, double b) {
        double r9172786 = b;
        double r9172787 = r9172786 * r9172786;
        double r9172788 = 4.0;
        double r9172789 = a;
        double r9172790 = 12.0;
        double r9172791 = r9172789 * r9172790;
        double r9172792 = r9172788 - r9172791;
        double r9172793 = r9172787 * r9172792;
        double r9172794 = r9172789 * r9172789;
        double r9172795 = r9172788 * r9172794;
        double r9172796 = r9172795 * r9172789;
        double r9172797 = r9172794 + r9172787;
        double r9172798 = sqrt(r9172797);
        double r9172799 = pow(r9172798, r9172788);
        double r9172800 = 1.0;
        double r9172801 = r9172799 - r9172800;
        double r9172802 = r9172796 + r9172801;
        double r9172803 = r9172795 + r9172802;
        double r9172804 = r9172793 + r9172803;
        return r9172804;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

2. Simplified 0.2

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.2

   \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(a \cdot a + b \cdot b\right) - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]

5. Applied associate-*l* 0.1

   \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(a \cdot a + b \cdot b\right)\right)} - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.1

   \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}\right) - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]

8. Applied cube-unmult 0.1

   \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]
9. Using strategy rm

10. Applied pow1 0.1

    \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3} - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]

11. Applied pow-prod-up 0.0

    \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left(\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\left(1 + 3\right)}} - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]

12. Simplified 0.0

    \[\leadsto \left(\left(a \cdot a\right) \cdot 4 + \left(\left(\left(a \cdot a\right) \cdot 4\right) \cdot a + \left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\color{blue}{4}} - 1\right)\right)\right) + \left(4 - a \cdot 12\right) \cdot \left(b \cdot b\right)\]

13. Final simplification 0.0

    \[\leadsto \left(b \cdot b\right) \cdot \left(4 - a \cdot 12\right) + \left(4 \cdot \left(a \cdot a\right) + \left(\left(4 \cdot \left(a \cdot a\right)\right) \cdot a + \left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right)\right)\right)\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))